Susan’s science class is performing an experiment that involves dropping objects from various heights, starting close to the ground and working upward to 8 feet. The function , where x represents the distance from the ground, represents the time it takes for the object Susan drops to hit the ground. The graph represents the function Identify some of the key features of the graph. That is, determine if the function is monotonically increasing or decreasing, state the end behavior, find the x- and y-intercepts, find the endpoint, and state the domain and the range of the graph (without considering the context).

1) The function is monotonically increasing

2) The end behavior of the function is x tends to infinity as t(x) tends infinity

3) The x and y -intercept is (0, 0)

4) The endpoint is (0, 0)

5) The domain is 0 ≤ x ≤ +∞

The range is 0 ≤ t(x) ≤ +∞

Step-by-step explanation:

1) The given function of the time for the object to hit the ground is t(x) = 1/4·√x

Where;

x = The distance from the ground

t = The time it takes for the object to hit the ground

Monotonically increasing function, we have;

A function that is continuous on [a, b] and it can be differentiated in the domain, (a, b) is monotonically increasing when df(x)/dx > 0 for all values of x in (a, b)

However, where df(x)/dx < 0 for all values of x in (a, b), the function is decreasing

Therefore, using an online tool, we have;

dt(x)/dx = d(1/4·√x)/dx = 1/8 × 1/√x

Therefore, the dt(x)/dx > 0, for 0 < x < +∞ and the function is monotonically increasing

2) The end behavior of the function as x tends to infinity, t(x) = 1/4·√x approaches infinity

3) From the end behavior, and the nature, of the function t(x) = 1/4·√x, where both variables are directly proportional, we have that the x and y -intercept =  (0, 0)

4) The endpoint is (0, 0) given that as t(x) tends to 0, x tends to 0

5) The domain is 0 ≤ x ≤ +∞

The range is 0 ≤ t(x) ≤ +∞.

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